MassWAssemblingP1OptV2.m

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function M=MassWAssemblingP1OptV2(nq,nme,me,areas,Tw)
% function M=MassWAssemblingP1OptV2(nq,nme,me,areas,Tw)
%   Assembly of the Weighted Mass Matrix using `P_1`-Lagrange finite elements
%   - OptV2 version (see report).
%
%   The Weighted Mass Matrix `\MasseF{w}` is given by 
%   ``\MasseF{w}_{i,j}=\int_\DOMH w(\q)\;\FoncBase_i(\q)\; \FoncBase_j(\q)\; d\q,\ \forall (i,j)\in{\ENS{1}{\nq}}^2``
%   where `\FoncBase_i` are `P_1`-Lagrange basis functions.
% Parameters:
%  nq: total number of nodes of the mesh, also denoted by `\nq`,
%  nme: total number of triangles, also denoted by `\nme`,
%  me: Connectivity array, `3\times\nme` array.<br/>
%  `\me(\jl,k)` is the storage index of the
%  `\jl`-th  vertex of the `k`-th triangle in the array `\q` of vertices coordinates, `\jl\in\{1,2,3\}` and
%       `k\in{\ENS{1}{\nme}}`.
%  areas: Array of areas, `1\times\nme` array. areas(k) is the area of the `k`-th triangle.
%  Tw: Array containing the values of `w` at the vertices,
%  `1\times\nq` array.<br/>
%  `Tw(i)=w(\q^i),` `\forall i\in\ENS{1}{\nq}`.
%
% Return values:
%  M: Global weighted mass matrix, `\nq\times\nq` sparse matrix.
%
% Example:
%  @verbatim 
%    Th=SquareMesh(10);
%    w=@(x,y) cos(x+y);
%    Tw=w(Th.q(1,:),Th.q(2,:));
%    Mw=MassWAssemblingP1OptV2(Th.nq,Th.nme,Th.me,Th.areas,Tw);
%  @endverbatim
% Copyright:
%   See \ref license
Ig = me([1 2 3 1 2 3 1 2 3],:);
Jg = me([1 1 1 2 2 2 3 3 3],:);
W=Tw(me).*(ones(3,1)*areas/30);
Kg=zeros(9,length(areas));
Kg(1,:)=3*W(1,:)+W(2,:)+W(3,:);
Kg(2,:)=W(1,:)+W(2,:)+W(3,:)/2;
Kg(3,:)=W(1,:)+W(2,:)/2+W(3,:);
Kg(5,:)=W(1,:)+3*W(2,:)+W(3,:);
Kg(6,:)=W(1,:)/2+W(2,:)+W(3,:);
Kg(9,:)=W(1,:)+W(2,:)+3*W(3,:);
Kg([4, 7, 8],:)=Kg([2, 3, 6],:);
M = sparse(Ig,Jg,Kg,nq,nq);